Certification Problem

Input (TPDB SRS_Standard/Secret_06_SRS/secr3)

The rewrite relation of the following TRS is considered.

Property / Task

Answer / Result

Proof (by matchbox @ termCOMP 2023)

1 Dependency Pair Transformation

1.1 Monotonic Reduction Pair Processor with Usable Rules

1.1.1 Dependency Graph Processor

The dependency pairs are split into 1 component.

• The 1^st component contains the pair

  c#(x1)	→	b#(x1)	(5)
  b#(b(b(x1)))	→	c#(b(x1))	(10)
  c#(x1)	→	a#(b(x1))	(6)
  a#(b(a(x1)))	→	c#(x1)	(11)
  a#(b(a(x1)))	→	b#(c(x1))	(12)

  1.1.1.1 Reduction Pair Processor with Usable Rules

  Using the matrix interpretations of dimension 1 with strict dimension 1 over the arctic semiring over the naturals

  [d(x1)]	=	-∞ · x1 + 29
  [c(x1)]	=	14 · x1 + 7
  [b(x1)]	=	7 · x1 + 0
  [a(x1)]	=	7 · x1 + 5
  [c#(x1)]	=	56 · x1 + 56
  [b#(x1)]	=	49 · x1 + -∞
  [a#(x1)]	=	49 · x1 + 56

  together with the usable rules

  a(b(a(x1)))	→	b(c(x1))	(1)
  b(b(b(x1)))	→	c(b(x1))	(2)
  c(x1)	→	a(b(x1))	(3)
  c(d(x1))	→	d(c(b(a(x1))))	(4)

  (w.r.t. the implicit argument filter of the reduction pair), the pairs

  c#(x1)	→	b#(x1)	(5)
  a#(b(a(x1)))	→	c#(x1)	(11)

  could be deleted.

  1.1.1.1.1 Dependency Graph Processor

  The dependency pairs are split into 1 component.

  • The 1^st component contains the pair

    b#(b(b(x1)))	→	c#(b(x1))	(10)
    c#(x1)	→	a#(b(x1))	(6)
    a#(b(a(x1)))	→	b#(c(x1))	(12)

    1.1.1.1.1.1 Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 3 with strict dimension 1 over the arctic semiring over the naturals

    [d(x1)]	=	-∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 2 -∞ 3
    [c(x1)]	=	5 4 0 9 5 1 13 9 5 · x1 + -∞ 7 11
    [b(x1)]	=	0 0 -∞ 3 4 0 9 5 0 · x1 + -∞ -∞ 7
    [a(x1)]	=	4 0 -∞ -∞ 0 0 9 5 4 · x1 + -∞ -∞ 11
    [c#(x1)]	=	14 10 5 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 15 -∞ -∞
    [b#(x1)]	=	8 6 -∞ -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 15 -∞ -∞
    [a#(x1)]	=	2 0 4 -∞ -∞ -∞ -∞ -∞ -∞ · x1 + 6 -∞ -∞

    together with the usable rules

    a(b(a(x1)))	→	b(c(x1))	(1)
    b(b(b(x1)))	→	c(b(x1))	(2)
    c(x1)	→	a(b(x1))	(3)
    c(d(x1))	→	d(c(b(a(x1))))	(4)

    (w.r.t. the implicit argument filter of the reduction pair), the pair

    c#(x1)

    →

    a#(b(x1))

    (6)

    could be deleted.

    1.1.1.1.1.1.1 Monotonic Reduction Pair Processor with Usable Rules

    Using the matrix interpretations of dimension 1 with strict dimension 1 over the rationals with delta = 1

    [d(x1)]	=	x1 + 0
    [c(x1)]	=	x1 + 0
    [b(x1)]	=	x1 + 0
    [a(x1)]	=	x1 + 0
    [c#(x1)]	=	x1 + 0
    [b#(x1)]	=	x1 + 1
    [a#(x1)]	=	x1 + 2

    together with the usable rules

    a(b(a(x1)))	→	b(c(x1))	(1)
    b(b(b(x1)))	→	c(b(x1))	(2)
    c(x1)	→	a(b(x1))	(3)
    c(d(x1))	→	d(c(b(a(x1))))	(4)

    (w.r.t. the implicit argument filter of the reduction pair), the pairs

    b#(b(b(x1)))	→	c#(b(x1))	(10)
    a#(b(a(x1)))	→	b#(c(x1))	(12)

    and no rules could be deleted.

    1.1.1.1.1.1.1.1 Dependency Graph Processor

    The dependency pairs are split into 0 components.